# Numerical boundary conditions for wide approximation finite difference

1. Jan 21, 2014

### amalak

Hi,

I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is:

u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2

or when discretized

u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2

In addition to dirichlet boundary conditions (which are not troubling me to implement), I have to implement numerical boundary conditions for

U-1 and UN+1

The problem I'm encountering is I'm not sure what to try for these numerical boundary conditions (as in, I haven't a clue as to what may work). I have the scheme set up without those conditions, but that's not what I want. The only time I know U-1 and UN+1 come up are with Neumann boundary conditions, which I don't have.

Any help or pointers would be immensely appreciated, thank you.

2. Jan 21, 2014

### AlephZero

3. Jan 22, 2014

### amalak

Thank you very much for your response. That method seems to make more sense, I think, but I've been instructed to use U-1 and UN+1, but thank you again.

4. Jan 22, 2014

### AlephZero

OK, so they probably want your numerical solution satisfy the differential equation at the boundary points $u_0$ and $u_N$, not just to satisfy the Dirichlet boundary conditions at the boundary points.

So, make finite difference approximations for the derivatives at $u_0$ using $u_{-1}, u_0,. u_1, \dots$ and plug them into the differential equation, and similarly at $u_N$. That will give you two more equations, so you have enough equations to solve for $u_{-1}, u_0,\dots, u_N, u_{N+1}$.